algorithms in percolation problem: how to identify and measure

Algorithms in Percolation

Problem: how to identify and measure cluster size

distribution

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Single-Cluster growth “Leath-Alexandrowicz method”

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P. L. Leath, Phys. Rev. B 14, 5046 (1976) Z. Alexandrowicz, Phys. Lett. A 80, 284 (1980).

Paul Leath Rutgers University

Leath-Alexandrowicz Algorithm

•  “Grow” clusters by adding sites one at a time from an initial seed

•  Two methods: –  “Breadth first” or “First-In-First-Out” (FIFO)

•  Requires making a list or queue –  “Depth first” or “Last-In-Last-Out” (LIFO)

•  Can be done using stack and recursion

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FIFO site percolation

Start with a seed site that is “wet”

Check neighbors in this order:

1

2

3

4

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Occupy a site with probability p

Orange = first shell


Unchecked sites queue:

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Occupy a site with probability p




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Make site “vacant” with probability 1-p



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Green = second shell


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Blue = third shell


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Blue = third shell


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Blue = third shell


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Blue = third shell


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Violet = fourth shell


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Violet = fourth shell


FIFO algorithm

“Pop” new growth site from queue For (neighbors = 1 to 4)

if (neighbor == unvisited) if (randomnumber < prob) neighbor = occupied “push” neighbor on queue else neighbor = vacant.

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LIFO site percolation

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Orange = growth from first neighbor of seed

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Put all growing sites on a stack

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Blue = growth from third neighbor of seed

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Violet = growth from fourth neighbor of seed

LIFO algorithm (can also use recursion)

Get new growth site from stack For (neighbors = 1 to 4)

if (neighbor == unvisited) if (randomnumber < prob) neighbor = occupied put neighbor on stack else neighbor = vacant.

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FIFO bond percolation

Red = seed “activated” or “wet” site

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Orange = first shell of bonds

Add bonds to dry sites with probability p

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Add bonds to dry sites with probability p

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Vacant bond with probability 1 - p

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Vacant bond with probability 1 - p

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Green = second shell of bonds

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Here we are concerned with the wet sites only and do not add bonds already wet sites

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Blue = third shell of bonds


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Violet = fourth shell of bonds


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Violet = fourth shell of bonds

The final object is a minimally spanning tree that connects to every wet site of the cluster

FIFO bond percolation (finding wetted sites)

“Pop” new growth site from queue For (neighbors = 1 to 4)

if (neighbor == unvisited) if (randomnumber < prob) neighbor = occupied “push” neighbor on queue else neighbor = vacant.

Identical to FIFO site perc. except for this line being taken out

Hoshen-Kopelman algorithm 1976

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J. Hoshen and R. Kopelman, Phys. Rev. B 14:3438 (1976).

Raoul Kopelman, University of Michigan

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Hoshen-Kopelman Algorithm

Look at last row of growing inteface.

Each color represents a connected cluster

(Bond percolation)

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Add bonds sequentially with probability p

(Bond percolation)

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(Bond percolation)

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(Bond percolation)

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(Bond percolation)

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(Bond percolation)

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(Bond percolation)

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(Bond percolation)

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Repeat!!!!

(Bond percolation)

Can simulate lattices as large as 100,000,000 x 100,000,000 this way!!!!!

Newman-Ziff algorithm 2000

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Newman-Ziff Algorithm

Start with an empty lattice, compute Q(Γ0) = Q0

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Randomly occupy a bond, compute Q(Γ1)

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Randomly occupy an unoccupied bond, compute Q(Γ2)

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And so on and compute Q(Γb) with b number of bonds

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Until all bonds are occupied, compute Q(ΓM)

M is the total number of bonds

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•  Any quantity as a function of p is computed as

where Qb = Q(Γb). •  Each sweep takes time of O(N)

0

! (1 )!( )!

Mb M b

bb

MQ p p Qb M b

−

=

= −−∑


M. E .J. Newman and R. M. Ziff, Phys. Rev. Letters 85, 4104 (2000)

•  Bonds are added one at a time, and a bookkeeping scheme is used to keep track of the cluster structure.

Mark Newman, U. Michigan

data structure

•  At each lattice site is a variable, ptr[i] •  If the ptr[i] > 0, it gives the position of another site on

the cluster (a link). •  If ptr[i] < 0, “i” is the root of the cluster, and |ptr[i]|

gives the number of sites belonging to the cluster.

procedure •  Initially, a random ordering of the bonds of the system is

made •  1. A bond is chosen randomly, and findroot is used to

find the root at each end (and the link paths are collapsed)

•  a) If the two ends belong to different clusters, the two are merged.

•  b) If both ends of the bond are in the same cluster, nothing is done.

findroot

-- findroot jumps from link to link until it gets to the root -- when the recursive calls “unwind”, they rename every link to point to the root

Before

After

merging

-- the root of the smaller cluster is linked to the root of the larger cluster, and the size is adjusted accordingly

convolution To go from the canonical (fixed number of bonds) to the grand-canonical (fixed occupancy p), one must convolve with a

binomial distribution:

Qn = quantity (such as mean size) for a system of fixed n Q(p) = same quantity for a fixed probability p

Example of Exact canonical-grand canonical: the crossing probability of a square system, up to 7 x 7 (from exact enumeration):

example: probability of wrapping a square torus in one direction but not

the other for all values of p •  L x L, L = 32, 64, 128, 256 •  Reaches maximum ≈ pc

•  Excellent convergence:

Easily find •  pc = 0.5927462..

p

“Percolation Threshold” Wikipedia page

ns = number of clusters of size s, at the critical threshold pc. τ = 187/91.

Used Cardys’ result for crossing of an annulus

to find size distribution

Simulation results – up to 2.5 x 1011 clusters up to size s = 1000.

Explosive growth in clusters created through a biased “Achlioptas” growth

process on a regular lattice

Achlioptas process •  Recently, Achlioptas, D’Sousa, and Spencer considered cluster growth

on random (Erdös-Rényi ) lattices by the so-called Achlioptas process: –  Pick two bonds –  Calculate weight = product of masses of the two clusters the bond connects –  Choose bond of lower weight

•  ER = Erdös-Rényi (regular percolation), BF = Bounded size rule, PR = product rule. C/n = maximum cluster size divided by the number of sites

•  They find “Explosive Growth” in the PR model.

Explosive growth

Achlioptas et al, Science 2009

Dimitris Achlioptas, UCSC

Raissa D’Sousa, UCD

Joel Spencer, NYU

Alfréd Rényi 1921-‐1970

My Erdös number is 2 by way of Mark Kac Tree form

Minimum spanning tree

Achlioptas processes on a regular (percolation) lattices

•  Define t = time = number of bonds added to

connect distinct clusters. •  Then, the number of clusters is n – t, where n

is the initial number of sites, since adding additional

Regular percolation transition at t/n = 1 – Nc/n = (7 – 3 √3)/2 =

0.9019

t/n

C/n

For laQce percolaRon, I find (1024x1024 laQce):

Product Rule (PR)

The SIR model on a square lattice

Susceptible-Infected-Recovered

With David de Souza and Tânia Tomé.

•  That is S → I with rate (1-c)Ineighbors/4 •  I →R with rate c

•  (I remains I for an exponentially distributed time)

algorithms in percolation problem: how to identify and measure

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